Probablility and The Fundamental Theorem of Poker

[lastcardcharlie]

Well I'd still like to make a couple of points, numeri, if I may:

(1) I'm sitting UTG pre-flop. No doubt I will get information about opponents' hole cards as the hand progresses, but right now I have no such information so does the FTOP apply to my initial decision whether to raise, call, or fold? In other words, why in this specific situation is one allowed to speak of being able to see opponents' hole cards but not future cards to be dealt?

(2) In a game of stud you've got spades showing but fold for a small bet in a large pot that a flush is likely to win. I conclude that your hole cards are not spades and, consequently, the probability of being dealt a spade next round is that little bit higher. In extreme cases this might affect my betting decision. I don't know, but it's a sincere question and I don't mean to be obtuse: don't stud players ever reason in this way?

[numeri]

Good questions.

(1) Yes, the FTOP does apply. Suppose you hold KK in a 10-handed hold'em game. Obviously, you're going to raise here, since KK is way ahead of the range of all the hand the other players at the table will play. Suppose then that a player in MP then 3-bets with AA (which you, of course, are unaware of).

Obviously, if you could go back to your previous decision, you might reconsider since KK is way behind AA. According to the FTOP, your choice to raise here was "incorrect". On the other hand, it is still the correct choice to make at the table, since you do not know that MP has AA.

It was an incorrect decision according to the FTOP, but a correct decision at the table. Again, the FTOP is not a theorm that helps us make decisions at the table.

(2) Apples and oranges. You're discussing a different question. At no point have I written that the type of decision you are describing is not correct. It has nothing to do with the FTOP. In that case, the FTOP deals with your decision in regards to the remaining players at the table. The fact that one player folded with spades showing will help you make your decision, but it has nothing to do with the FTOP.

Example: (My experience is hold'em, so I'll use that.)

I raise A[img]/images/graemlins/spade.gif[/img]K[img]/images/graemlins/spade.gif[/img] UTG and get cold-called by a loose-passive MP player and get another call from the BB, who is a 2+2er.

Flop (3 players) 9[img]/images/graemlins/spade.gif[/img] 5[img]/images/graemlins/spade.gif[/img] 2[img]/images/graemlins/diamond.gif[/img]
BB checks, I bet, MP calls, BB folds.

Since BB folded here, I can be certain that he did not have two spades, so I would generally give myself about 8 outs for the flush (since MP might have some), plus a standard discounted 3 outs for my overcards. (Of course, I may also be ahead.)

My point, however, is that the information from BB has anything to do with the FTOP. It all refers to how I make my decisions after this point - i.e. heads-up versus MP on the turn. I only care about MP's cards at that point. The fact that there are more spades available helps me narrow down MP's hand and make a better decision according to the FTOP - since I can narrow his range to overcards, a pocket pair, a flush draw, and possibly a gut-shot.

Turn: (2 players) 3[img]/images/graemlins/heart.gif[/img]

Suppose MP has 6[img]/images/graemlins/diamond.gif[/img]6[img]/images/graemlins/heart.gif[/img]. He will call down. My decision to bet (as I tend to do here to extract value from any overcards and draws) would then be a "mistake" according to the FTOP, but correct (IMO) given the limited information I have.

I'm rambling now, but it really seems like you are mixing two concepts up. You keep referring to how knowing what cards are out will help you make your decision - that is probability theory. Knowing that there is a higher chance to have more [img]/images/graemlins/spade.gif[/img]s out helps me calculate the odds that I am ahead. It does not, however, give me any information about MP's hand and help with the FTOP.

I'm not sure if this helped at all or just confused things. It seems clear to me that you are trying to apply the FTOP in an incorrect manner. The points you are making are not incorrect - they are just about a different aspect of the game.

[four-flush]

[ QUOTE ]
In response to four-flush's post, I am not here for one moment suggesting that mathematical probability is flawed

[/ QUOTE ]

Yes you are because you said, "Probability is to a large extent not a measure of reality."

[ QUOTE ]
What I am questioning is its use as foundation for a prescriptive theory of how to play good poker.

[/ QUOTE ]

Just remember that most of the skill used to win at poker is not mathematical, suffice it to say that probabilility has existed for half a millenium, it has been applied successfully to poker in both theory and practice for decades. You are the only person I have ever heard of who contests the method of application of probability to poker. I don't believe the leading authorities on the mathematics of poker (Caro for eg.) would agree with your contentions.

[ QUOTE ]
To return to the meaning of probability, I repeat that it is a measure of what one knows about reality rather than a measure of reality itself

[/ QUOTE ]

No, the meaning of probability is this: a measure of how likely it is that some event will occur.



One cannot discount something that is incontestable. I suggest that you write to the author of the `FTOP' and discuss the matter with him, or with a professor of mathematics and statistics. I think I'll leave it at that and let you talk with others in the forum to get their perspective on this. There's not much more I can add, other than what I have already said.
[img]/images/graemlins/diamond.gif[/img] [img]/images/graemlins/club.gif[/img] [img]/images/graemlins/spade.gif[/img]

[numeri]

[ QUOTE ]
I suggest that you write to the author of the `FTOP' and discuss the matter with him, or with a professor of mathematics and statistics.

[/ QUOTE ]
FWIW, I satisfy these qualifications. I'm not an expert by any means, but I do have some experience in the field.

[lastcardcharlie]

[ QUOTE ]

It was an incorrect decision according to the FTOP, but a correct decision at the table. Again, the FTOP is not a theorem that helps us make decisions at the table.


[/ QUOTE ]

I think I do understand this point. Raising with KK was the right play given the information available to the UTG at the time, but was a "mistake" (Sklansky calls it "special kind of mistake") according to the FTOP. I think it is important to note that natural language becomes a little distorted here.

That the FTOP is "not a theorem that helps us make decisions at the table" is, as you have observed, a view that is supported by a reading of the strategy forums here - they are not discussing the FTOP. That it is not helpful was also one of the claims in my original post. This seems, however, to be a very strange reading of TOP. The view that the FTOP is indeed a theorem that helps us make decisions at the table is, I would argue, clearly the major theme of TOP, and numerous explicit examples are given.

What I still don't understand is the essential difference between the following two statements:

(A) My opponent had AA so I wish I hadn't raised with KK even though I know it was still the right thing to do;

(B) My opponent had QQ so I wish I hadn't raised pre-flop with KK because the flop was QQx, even though I know it was still the right thing to do.

Now, if I understand correctly, you have explained that the difference consists in the possibility of gaining information about hole cards on the same betting round, and this is what I just don't get. So what if you do? You might know that you've made a "special kind of mistake" on the same betting round rather than on the next, but so what? Why does this matter?

Suppose the wording "if you could see all your opponents' cards" becomes true for a card cheat. Now suppose two cheats are playing poker and one can see all the hole cards but the other is a better cheat and can also see the flop, turn, river. The first cheat is, according to the FTOP, playing perfect poker but still gets beaten. Why?

Another observation I'd like to make is that probably most of the examples in TOP are about gain/loss according to what an opponent does in relation to one's own hole cards. Now I'm wondering if, from my perspective, there is not some ontological difference between "if I could see my opponents' cards" and "if opponents could see my cards". Sklansky treats these as perfectly converse, which they certainly seem to be, but on a brief re-reading of TOP the latter does seem to figure far more in his examples.

[numeri]

I"m sorry, but I'm out of answers. We're talking about two different things. Example (A) is actually possible to find out and change your play the rest of the hand (say if the player only 3-bets with AA or KK), but example (B) is impossible to know and therefore imaterial. Who cares? You can never know that information, so it's pointless to discuss.

[Daisydog]

Most of us would agree that what makes a poker player good is the ability to make good decisions based on available information (which is incomplete). In Texas Hold em, when you are making a decision, information is incomplete for at least two reasons:
1. uncertainty in the board cards to come on later streets
2. uncertainty in your opponent's hole cards.

Both of these can be described with probability distributions.

The FTOP states,". . . every time you play your hand the same way you would have played it if you could see all of their cards, they lose . . ."

Of course, the FTOP doesn't literally mean every time. It is saying "they lose" from an EV perspective. The board cards on a later street may come up in such a way that your opponent sucks out on you. But the board cards follow a probability distribution, and from an EV perspective your opponent loses.

Similarly, your opponent's hole cards also follow a probability distribution. So there are plays that may be bad given an opponents exact hole cards, but may be good from an EV perspective given the opponent's hole card distribution. For example, it might be a good decision to raise UTG with KK, even if a MP player has AA. We don't know the MP player has AA and raising with KK is +EV given the hole card distributions.

I think what lastcardcharlie is trying to say is that the FTOP deals with the probability distributions of future board cards, but not the probability distribution of opponent's hole cards. Therefore, it can tell us that a decision was good even if an opponent sucks out on us. However it will not allow us to say that we made a good decision when we lose because the opponent's hole cards just happen to be on the extreme high end of their distribution.

In poker we make decisions with incomplete information. This incomplete information lends itself to being described by probability distributions and EV analysis. In Texas Hold em the incomplete information stems from uncertainty in future board cards as well as from uncertainty in opponents hole cards. Decisions can be good even if board cards OR opponents hole cards turn out to be unfavorable. The FTOP appears to deal only with one aspect of the incomplete information (board cards) but not the other (opponent's hole cards). Does this mean it is wrong? No. It's correct, it just doesn't deal with uncertainty in opponent's hole cards.

In my mind, the most fundamental thing about good poker playing is making good decisions given all AVAILABLE information. Exact hole cards are not available, but hole card distributions are to those who are observant. Perhaps the FTOP ought to incorporate the idea that decisions can be good even if they turn out to be bad once the hole cards are known.

Thanks, lastcardcharlie, for a very interesting topic. I look forward to future discussion.